Differential Equations: An Introduction to Modern Methods and Applications (Hardcover, 3, Revised)

Chapter 1: Introduction

1.1 Mathematical Models and Solutions

1.2 Qualitative Methods: Phase Lines and Direction Fields

1.3 Definitions, Classification, and Terminology

Chapter 2: First Order Differential Equations

2.1 Separable Equations

2.2 Linear Equations: Method of Integrating Factors

2.3 Modeling with First Order Equations

2.4 Differences Between Linear and Nonlinear Equations

2.5 Autonomous Equations and Population Dynamics

2.6 Exact Equations and Integrating Factors

2.7 Substitution Methods

Projects

2.P.1 Harvesting a Renewable Resource

2.P.2 A Mathematical Model of a Groundwater Contaminant Source

2.P.3 Monte Carlo Option Pricing: Pricing Financial Options by Flipping a Coin

Chapter 3: Systems of Two First Order Equations

3.1 Systems of Two Linear Algebraic Equations

3.2 Systems of Two First Order Linear Differential Equations

3.3 Homogeneous Linear Systems with Constant Coefficients

3.4 Complex Eigenvalues

3.5 Repeated Eigenvalues

3.6 A Brief Introduction to Nonlinear Systems

Projects

3.P.1 Estimating Rate Constants for an Open Two-Compartment Model

3.P.2 A Blood-Brain Pharmacokinetic Model

Chapter 4: Second Order Linear Equations

4.1 Definitions and Examples

4.2 Theory of Second Order Linear Homogeneous Equations

4.3 Linear Homogeneous Equations with Constant Coefficients

4.4 Mechanical and Electrical Vibrations

4.5 Nonhomogeneous Equations; Method of Undetermined Coefficients

4.6 Forced Vibrations, Frequency Response, and Resonance

4.7 Variation of Parameters

Projects

4.P.1 A Vibration Insulation Problem

4.P.2 Linearization of a Nonlinear Mechanical System

4.P.3 A Spring-Mass Event Problem

4.P.4 Euler-Lagrange Equations

Chapter 5: The Laplace Transform

5.1 Definition of the Laplace Transform

5.2 Properties of the Laplace Transform

5.3 The Inverse Laplace Transform

5.4 Solving Differential Equations with Laplace Transforms

5.5 Discontinuous Functions and Periodic Functions

5.6 Differential Equations with Discontinuous Forcing Functions

5.7 Impulse Functions

5.8 Convolution Integrals and Their Applications

5.9 Linear Systems and Feedback Control

Projects

5.P.1 An Electric Circuit Problem

5.P.2 The Watt Governor, Feedback Control, and Stability

Chapter 6: Systems of First Order Linear Equations

6.1 Definitions and Examples

6.2 Basic Theory of First Order Linear Systems

6.3 Homogeneous Linear Systems with Constant Coefficients

6.4 Nondefective Matrices with Complex Eigenvalues

6.5 Fundamental Matrices and the Exponential of a Matrix

6.6 Nonhomogeneous Linear Systems

6.7 Defective Matrices

Projects

6.P.1 Earthquakes and Tall Buildings

6.P.2 Controlling a Spring-Mass System to Equilibrium

Chapter 7: Nonlinear Differential Equations and Stability

7.1 Autonomous Systems and Stability

7.2 Almost Linear Systems

7.3 Competing Species

7.4 Predator-Prey Equations

7.5 Periodic Solutions and Limit Cycles

7.6 Chaos and Strange Attractors: The Lorenz Equations

Projects

7.P.1 Modeling of Epidemics

7.P.2 Harvesting in a Competitive Environment

7.P.3 The Rossler System

Chapter 8: Numerical Methods

8.1 Numerical Approximations: Euler’s Method

8.2 Accuracy of Numerical Methods

8.3 Improved Euler and Runge-Kutta Methods

8.4 Numerical Methods for Systems of First Order Equations

Projects

8.P.1 Designing a Drip Dispenser for a Hydrology Experiment

8.P.2 Monte Carlo Option Pricing: Pricing Financial Option by Flipping a Coin

Chapter 9: Series Solutions of Second order Equations

9.1 Review of Power Series

9.2 Series Solutions Near an Ordinary Point, Part I

9.3 Series Solutions Near an Ordinary Point, Part II

9.4 Regular Singular Points

9.5 Series Solutions Near a Regular Singular Point, Part I

9.6 Series Solutions Near a Regular Singular Point, Part II

9.7 Bessel’s Equation

Projects

9.P.1 Diffraction Through a Circular Aperature

9.P.2 Hermite Polynomials and the Quantum Mechanical Harmonic Oscillator

9.P.3 Perturbation Methods

Chapter 10: Orthogonal Functions, Fourier Series and Boundary-Value Problems

10.1 Orthogonal Families in the Space PC [a,b]

10.2 Fourier Series

10.3 Elementary Two-Point Boundary Value Problems

10.4 General Sturm-Liouville Boundary Value Problems

10.5 Generalized Fourier Series and Eigenfunction Expansions

10.6 Singular Boundary Value Problems

10.7 Convergence Issues

Chapter 11: Elementary Partial Differential Equations

11.1 Terminology

11.2 Heat Conduction in a Rod—Homogeneous Case

11.3 Heat Conduction in a Rod—Nonhomogeneous Case

11.4 Wave Equation—Vibrations of an Elastic String

11.5 Wave Equation—Vibrations of a Circular Membrane

11.6 Laplace Equation

Projects

11.P.1 Estimating the Diffusion Coefficient in the Heat Equation

11.P.2 The Transmission Line Problem

11.P.3 Solving Poisson’s Equation by Finite Differences

11.P.4 Dynamic Behavior of a Hanging Cable

11.P.5 Advection Dispersion: A Model for Solute Transport in Saturated Porous Media

11.P.6 Fisher’s Equation for Population Growth and Dispersion

Appendices

11.A Derivation of the Heat Equation

11.B Derivation of the Wave Equation

A: Matrices and Linear Algebra

A.1 Matrices

A.2 Systems of Linear Algebraic Equations, Linear Independence, and Rank

A.3 Determinates and Inverses

A.4 The Eigenvalue Problem

B: Complex Variables

Answers to Selected Problems

References

Index